First-order models · Definition

Logistic model

differential equationsapplications

Before you start you need to know

Exponential growth assumes unlimited resources; the logistic model corrects this with a brake that appears as the population approaches what the environment can support.

Definition

Model of a population \(P\) with intrinsic rate \(r>0\), carrying capacity \(K>0\) and initial population \(P_0\): it grows almost exponentially at first and levels off at \(K\).

\[P'=rP\left(1-\frac{P}{K}\right)\;\Longrightarrow\; P(t)=\frac{K}{1+\left(\frac{K}{P_0}-1\right)e^{-rt}}\]

The model starts from \(P'=rP\) and multiplies it by the factor \(1-P/K\), which is almost \(1\) when \(P\) is small and vanishes when \(P=K\). It is autonomous and separable: with partial fractions, \(\int\frac{dP}{P(1-P/K)}=\int r\,dt\) leads to \(P(t)=K\big/\big(1+(K/P_0-1)e^{-rt}\big)\).

The equilibria are \(P=0\) and \(P=K\). If \(0<P_0<K\), the solution grows toward \(K\); if \(P_0>K\), it decreases toward \(K\). So \(K\) attracts every positive population and \(0\) repels it, which is why \(K\) is the value the population settles at.

The S-shaped curve grows fastest at \(P=K/2\), where \(P'=rK/4\) and the graph changes from concave up to concave down. When \(P\ll K\) it behaves like \(y'=ry\), and the time to reach \(K/2\) is \(t=\ln(K/P_0-1)/r\).

Worked example: with \(r=0.5\), \(K=1000\) and \(P_0=100\) the population reaches \(500\) at \(t=2\ln9\approx4.4\). A typical mistake is reading \(r\) as the observed rate: the actual per-capita rate is \(r(1-P/K)\), which shrinks as \(P\) grows.

Think about it

In \(P'=3P-P^2\), what are \(r\) and \(K\)?

Factoring, \(P'=3P\left(1-\tfrac P3\right)\): \(r=3\) and \(K=3\).

Exercises 10

  1. Find the equilibrium points of \(P'=0.5\,P\left(1-\frac{P}{100}\right)\).
  2. The equation \(P'=0.2\,P-0.002\,P^2\) is logistic. What is its carrying capacity \(K\)?
  3. In \(P'=0.03\,P\left(1-\frac{P}{500}\right)\), what are \(r\) and \(K\)?
  4. In the logistic model, if \(0<P<K\), how does \(P\) evolve?
  5. For \(P'=0.4\,P\left(1-\frac{P}{200}\right)\), at what population is the growth rate \(P'\) largest?
  6. For \(P'=P\left(1-\frac{P}{10}\right)\) with \(P(0)=1\), the solution is \(P(t)=\dfrac{10}{1+9e^{-t}}\). What is \(P(\ln 9)\)?
  7. With \(r=0.1\) and \(K=100\), a population \(P=300\) exceeds the carrying capacity. What is \(P'\) at that moment?
  8. With \(r=\tfrac12\), \(K=1000\) and \(P_0=100\), at what time does \(P\) reach \(K/2\)?
  9. Two logistic populations have the same \(r\) and \(K=100\), but \(P_0=5\) and \(P_0=60\). What happens in the long run?
  10. For \(0<P<K\) we have \(P''=r\left(1-\tfrac{2P}{K}\right)P'\). When is the graph of \(P(t)\) concave up?

References

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §2.5
  • Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §3.2
  • Steven H. Strogatz, Nonlinear Dynamics and Chaos, 2nd edition, Westview Press, 2015. §2.3